12,310 research outputs found
Positive degree and arithmetic bigness
We establish, for a generically big Hermitian line bundle, the convergence of
truncated Harder-Narasimhan polygons and the uniform continuity of the limit.
As applications, we prove a conjecture of Moriwaki asserting that the
arithmetic volume function is actually a limit instead of a sup-limit, and we
show how to compute the asymptotic polygon of a Hermitian line bundle, by using
the arithmetic volume function
Convergence of Harder-Narasimhan polygons
We establish in this article convergence results of normalized
Harder-Narasimhan polygons both in geometric and in arithmetic frameworks by
introducing the Harder-Narasimhan filtration indexed by and the
associated Borel probability measure
Finding largest small polygons with GloptiPoly
A small polygon is a convex polygon of unit diameter. We are interested in
small polygons which have the largest area for a given number of vertices .
Many instances are already solved in the literature, namely for all odd ,
and for and 8. Thus, for even , instances of this problem
remain open. Finding those largest small polygons can be formulated as
nonconvex quadratic programming problems which can challenge state-of-the-art
global optimization algorithms. We show that a recently developed technique for
global polynomial optimization, based on a semidefinite programming approach to
the generalized problem of moments and implemented in the public-domain Matlab
package GloptiPoly, can successfully find largest small polygons for and
. Therefore this significantly improves existing results in the domain.
When coupled with accurate convex conic solvers, GloptiPoly can provide
numerical guarantees of global optimality, as well as rigorous guarantees
relying on interval arithmetic
Mathematical Magic: A Study of Number Puzzles
Within this paper, we will briefly review the history of a collection of number puzzles which take the shape of squares, polygons, and polyhedra in both modular and nonmodular arithmetic. Among other results, we develop construction techniques for solutions of both Modulo and regular Magic Squares. For other polygons in nonmodular arithmetic, specifically of order 3, we present a proof of why there are only four Magic Triangles using linear algebra, disprove the existence of the Magic Tetrahedron in two ways, and utilizing the infamous 3-SUM combinatorics problem we disprove the existence of the Magic Octahedron
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